![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > uspgrushgr | Structured version Visualization version GIF version |
Description: A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.) |
Ref | Expression |
---|---|
uspgrushgr | ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2778 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | isuspgr 26518 | . . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
4 | ssrab2 3908 | . . . . 5 ⊢ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}) | |
5 | f1ss 6358 | . . . . 5 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})) | |
6 | 4, 5 | mpan2 681 | . . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})) |
7 | 3, 6 | syl6bi 245 | . . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
8 | 1, 2 | isushgr 26426 | . . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
9 | 7, 8 | sylibrd 251 | . 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)) |
10 | 9 | pm2.43i 52 | 1 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 {crab 3094 ∖ cdif 3789 ⊆ wss 3792 ∅c0 4141 𝒫 cpw 4379 {csn 4398 class class class wbr 4888 dom cdm 5357 –1-1→wf1 6134 ‘cfv 6137 ≤ cle 10414 2c2 11435 ♯chash 13441 Vtxcvtx 26361 iEdgciedg 26362 USHGraphcushgr 26422 USPGraphcuspgr 26514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-nul 5027 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fv 6145 df-ushgr 26424 df-uspgr 26516 |
This theorem is referenced by: uspgrupgrushgr 26543 usgredgedg 26593 vtxdusgrfvedg 26856 1loopgrvd2 26868 isomuspgr 42761 |
Copyright terms: Public domain | W3C validator |